It is legal here in sd as long as no lights are used. We hunt them in the spring as soon as the ice comes off the rivers. We use 22s and shoot them in the head. You are correct in saying they do not float for long! I don't know if a fishing set up would hold up to them. They will most likely break or BITE off your arrow if you don't hit them in the neck or head. Good luck and let us know if you get any. I might try it with a bow once myself.
PS: They are most active in the water during early morning and right before dark. You can catch them sleeping in the sun on a bank during the day.